Flat left-invariant pseudo-Riemannian metrics on unimodular Lie groups

We observe that, contrary to the Lorentzian case, there exist flat left-invariant pseudo-Riemannian metrics on a non-unimodular Lie group such that the center of its Lie algebra z (g) is degenerate. If the connected Lie group G is unimodular, then we show that if G admits a flat left-invariant pseudo-Riemmanian metric of signature (2, n-2) such that z (g) is degenerate, then ᵦ=0 for any z z (g) z (g) ^, where is the Levi-Civita connection of (G, ). Using this fact, we show that its Lie algebra is obtained by the double extension process from a flat Lorentzian unimodular Lie algebra. As examples, we give a classification of these Lie algebras in dimension 4. We also give a generalization of Milnor’s theorem to any flat left-invariant pseudo-Riemannian metric such that g, g is Euclidean.